Magnetic position sensor

ABSTRACT

A method of assembling a position sensing arrangement for sensing the position of a revolute joint of an articulated structure. The position sensing arrangement comprises a magnetic sensor assembly and a disc having a first magnetic ring with j magnetic pole pairs and a second magnetic ring with k magnetic pole pairs. A boundary of the disc is constrained by the articulated structure. The method comprises: determining a number of pole pairs of the first magnetic ring to be an integer p such that the first magnetic ring is separated from the constrained boundary by at least the magnetic sensor assembly; determining a number of pole pairs of the second magnetic ring to be an integer q such that the second magnetic ring is separated from the first magnetic ring by a predetermined distance; and if p and q are co-prime: selecting j to be p and k to be q; and assembling the position sensing arrangement by mounting the disc to the articulated structure such that both the disc and the revolute joint are permitted to rotate about the same axis.

BACKGROUND

In applications involving articulated structures, it is often desirableto determine the position of the distal end of the most distal link ofthe articulated structure. This can be achieved by sensing the positionof each link relative to the last along the articulated structure, fromits base to the most distal link. This series of measurements can beused in combination with the known layout of the articulated structureto determine the position of the distal end of the most distal linkrelative to the base. Rotary position sensors are used to sense relativerotation between links. Linear position sensors are used to senserelative longitudinal motion between links.

Hall effect magnetic sensors are commonly used to sense relative motionbetween links. In a typical rotary position sensor, a ring has a set ofalternating magnetic poles arranged around it. A sensor interacts withthe ring, and is located so that the magnetic poles move past the sensoras the rotation that is desired to be sensed takes place. For example,the ring could be attached about a shaft and the sensor could beattached to a housing within which the shaft rotates. The sensor detectschanges in magnetic polarity as the poles move past the sensor. Bycounting the number of changes in polarity the amount of rotation from areference position can be sensed. To sense the direction of rotation twosuch pairs of rings and sensors can be provided, and arranged so thatone sensor detects magnetic transitions of its ring at rotationpositions that are offset from the positions where the other sensordetects magnetic transitions of its ring. By considering the relativetiming of transitions detected by each sensor the direction of rotationcan be sensed.

The field of robotics utilises articulated structures as robot arms.Accurate position sensing is important for robot arms in order to ensuretheir end effectors are manipulated precisely as intended. The largerthe magnetic rings of the position sensor, the more accurately therelative rotation of two links of the robot arm is sensed. However, insome robotics applications, for example in the field of surgicalrobotics, it is desirable for the position sensors to be very compact tofit within the available space and to minimise the weight that they addto the arm.

Thus, there is a need for an improved position sensor which balances thecompeting requirements of accuracy and compactness.

SUMMARY OF THE INVENTION

According to an aspect of the invention, there is provided a method ofassembling a position sensing arrangement for sensing the position of arevolute joint of an articulated structure, the position sensingarrangement comprising a magnetic sensor assembly and a disc having afirst magnetic ring with j magnetic pole pairs and a second magneticring with k magnetic pole pairs, a boundary of the disc beingconstrained by the articulated structure, the method comprising:determining a number of pole pairs of the first magnetic ring to be aninteger p such that the first magnetic ring is separated from theconstrained boundary by at least the magnetic sensor assembly;determining a number of pole pairs of the second magnetic ring to be aninteger q such that the second magnetic ring is separated from the firstmagnetic ring by a predetermined distance; and if p and q are co-prime:selecting j to be p and k to be q; and assembling the position sensingarrangement by mounting the disc to the articulated structure such thatboth the disc and the revolute joint are permitted to rotate about thesame axis.

If p and q are not co-prime, the method may comprise: iterativelydetermining a further value q for the number of pole pairs of the secondmagnetic ring such that the difference between p and q increments by oneeach iteration; and for each iteration, if p and q are co-prime:selecting p to be a bound of a range of numbers of pole pairs of thefirst magnetic ring, and selecting q to be a bound of a range of numbersof pole pairs of the second magnetic ring; identifying one or more otherco-prime pair of numbers p′, q′, where: p′ is in the range of numbers ofpole pairs of the first magnetic ring, and q′ is in the range of numbersof pole pairs of the second magnetic ring, and for that p′, q′ pair, thesecond magnetic ring is separated from the first magnetic ring by atleast the predetermined distance; selecting the identified p′, q′ pairwhich has the largest value of p′; selecting j to be the selected p′ andk to be the selected q′; and assembling the position sensing arrangementby mounting the disc to the articulated structure such that both thedisc and the revolute joint are permitted to rotate about the same axis.

If p and q are not co-prime, the method may comprise: iterativelydetermining a further value q for the number of pole pairs of the secondmagnetic ring such that the difference between p and q increments by oneeach iteration; and for each iteration, if p and q are co-prime:selecting p to be a bound of a range of numbers of pole pairs of thefirst magnetic ring, and selecting q to be a bound of a range of numbersof pole pairs of the second magnetic ring; identifying one or more otherco-prime pair of numbers p′, q′, where: p′ is in the range of numbers ofpole pairs of the first magnetic ring, and q′ is in the range of numbersof pole pairs of the second magnetic ring, and for that p′, q′ pair, thesecond magnetic ring is separated from the first magnetic ring by atleast the predetermined distance; selecting the identified p′, q′ pairwhich has the largest value of q′; selecting j to be the selected p′ andk to be the selected q′; and assembling the position sensing arrangementby mounting the disc to the articulated structure such that both thedisc and the revolute joint are permitted to rotate about the same axis.

If p and q are not co-prime, the method may comprise: iterativelydetermining a further value q for the number of pole pairs of the secondmagnetic ring such that the difference between p and q increments by oneeach iteration; and for each iteration, if p and q are co-prime:selecting p to be a bound of a range of numbers of pole pairs of thefirst magnetic ring, and selecting q to be a bound of a range of numbersof pole pairs of the second magnetic ring; identifying one or more otherco-prime pair of numbers p′, q′, where: p′ is in the range of numbers ofpole pairs of the first magnetic ring, and q′ is in the range of numbersof pole pairs of the second magnetic ring, and for that p′, q′ pair, thesecond magnetic ring is separated from the first magnetic ring by atleast the predetermined distance; selecting the identified p′, q′ pairwhich has the smallest value of q′; selecting j to be the selected p′and k to be the selected q′; and assembling the position sensingarrangement by mounting the disc to the articulated structure such thatboth the disc and the revolute joint are permitted to rotate about thesame axis.

If p and q are not co-prime, the method may comprise: iterativelydetermining a further value q for the number of pole pairs of the secondmagnetic ring such that the difference between p and q increments by oneeach iteration; and for each iteration, if p and q are co-prime:selecting p to be a bound of a range of numbers of pole pairs of thefirst magnetic ring, and selecting q to be a bound of a range of numbersof pole pairs of the second magnetic ring; identifying one or more otherco-prime pair of numbers p′, q′, where: p′ is in the range of numbers ofpole pairs of the first magnetic ring and q′ is in the range of numbersof pole pairs of the second magnetic ring, and for that p′, q′ pair, thesecond magnetic ring is separated from the first magnetic ring by atleast the predetermined distance; selecting the identified p′, q′ pairwhich has the smallest value of |p′-q′|; selecting j to be the selectedp′ and k to be the selected q′; and assembling the position sensingarrangement by mounting the disc to the articulated structure such thatboth the disc and the revolute joint are permitted to rotate about thesame axis.

The first magnetic ring and the second magnetic ring may be concentric,the first magnetic ring being inside the second magnetic ring, theconstrained boundary being the inner radial boundary of the disc.

The method may comprise iteratively determining a further value q byincrementing q by 1 each iteration, wherein p′>p and q′<q, and whereinfor each iteration, if p and q are co-prime, the method comprisesselecting p to be the lower bound of a range of numbers of pole pairs ofthe first magnetic ring, and selecting q to be the upper bound of arange of numbers of pole pairs of the second magnetic ring.

The first magnetic ring and the second magnetic ring may be concentric,the first magnetic ring being outside the second magnetic ring, theconstrained boundary being the outer radial boundary of the disc.

The method may comprise iteratively determining a further value q bydecrementing q by 1 each iteration, wherein p′<p and q′>q, and whereinfor each iteration, if p and q are co-prime, the method comprisesselecting p to be the upper bound of a range of numbers of pole pairs ofthe first magnetic ring, and selecting q to be the lower bound of arange of numbers of pole pairs of the second magnetic ring.

If p and q are not co-prime, the method may comprise: iterativelydetermining a further value q for the number of pole pairs of the secondmagnetic ring such that the difference between p and q increments by oneeach iteration; and for each iteration, if p and q are co-prime:selecting p to be a bound of a range of numbers of pole pairs of thefirst magnetic ring, and selecting q to be a bound of a range of numbersof pole pairs of the second magnetic ring; identifying one or more otherco-prime pair of numbers p′, q′, where: p′ is in the range of numbers ofpole pairs of the first magnetic ring, and q′ is in the range of numbersof pole pairs of the second magnetic ring, and for that p′, q′ pair, thesecond magnetic ring is separated from the first magnetic ring by atleast the predetermined distance; selecting an identified p′, q′ pairdependent on the maximum angle of rotation of the revolute joint to bedetected by the position sensing arrangement; selecting j to be theselected p′ and k to be the selected q′; and assembling the positionsensing arrangement by mounting the disc to the articulated structuresuch that both the disc and the revolute joint are permitted to rotateabout the same axis.

The disc may further comprise a third magnetic ring with I pole pairs,and the method may further comprise: determining a number of pole pairsof the third magnetic ring to be an integer s such that the thirdmagnetic ring is separated from the second magnetic ring by a furtherpredetermined distance; and if p, q and s are co-prime: selecting j tobe p, k to be q, and I to be s; and assembling the position sensingarrangement by mounting the disc to the articulated structure such thatboth the disc and the revolute joint are permitted to rotate about thesame axis.

According to a further aspect of the invention, there is provided aposition sensing arrangement comprising: a first magnetic array having jmagnetic pole pairs; a second magnetic array having k magnetic polepairs, the second magnetic array being immovable relative to the firstmagnetic array; and a magnetic sensor assembly for detecting therelative position of the magnetic sensor assembly and the first andsecond magnetic arrays; wherein j and k are co-prime and |j-k|>1.

Suitably, |j-|>7.

The position sensing arrangement may be mounted to an articulatedstructure, wherein the first magnetic array is a first magnetic ring,the second magnetic array is a second magnetic ring, and the positionsensing arrangement is for sensing position of a revolute joint of thearticulated structure, the first magnetic ring and the second magneticring being mounted to the articulated structure such that the firstmagnetic ring, the second magnetic ring and the revolute joint are allpermitted to rotate about the same axis.

The first and second magnetic rings may both be disposed on a disc. Thefirst and second magnetic rings may be disposed on the same surface ofthe disc or opposing surfaces of the disc. The first and second magneticrings may be separated along the axis. The second magnetic ring may beradially separated from the first magnetic ring by a predetermineddistance.

The predetermined distance may be at least the length of a magnetic polepair.

The magnetic sensor assembly may comprise a first magnetic sensor arraydisposed over the first magnetic array and a second magnetic sensorarray disposed over the second magnetic array, adjacent sensors of eachof the first and second magnetic sensor arrays being separated by aquarter the length of a magnetic pole pair.

The first magnetic sensor array may have a radial extent less than theradial extent of the first magnetic ring, and the second magnetic sensorarray may have a radial extent less than the radial extent of the secondmagnetic ring.

Each of the first and second magnetic sensor arrays may be rectilinear.Each of the first and second magnetic sensor arrays may comprise foursensors. Either or both of the first and second magnetic sensor arraysmay be arranged in a circular configuration.

The position sensing arrangement may further comprise a third magneticring having I magnetic pole pairs, the third magnetic ring beingimmovable relative to the first and second magnetic rings, the thirdmagnetic ring being mounted to the articulated structure such that it ispermitted to rotate about the same axis as the first and second magneticrings, wherein j, k and I are co-prime and |l-k|>1 and |l-j|>1.

BRIEF DESCRIPTION OF THE FIGURES

The present invention will now be described by way of example withreference to the accompanying drawings. In the drawings:

FIG. 1 illustrates a general representation of a shaft equipped with aposition sensing arrangement;

FIG. 2 illustrates the dimensions of a disc 3 of FIG. 1;

FIG. 3 illustrates a portion of a magnetic ring of FIG. 1;

FIG. 4 illustrates a magnetic sensor array of FIG. 1;

FIG. 5 is a graph illustrating theoretical and actual position sensorreadings;

FIG. 6 is a flowchart illustrating a method of determining pole pairnumbers for two magnetic rings when the inner radial boundary islimiting;

FIG. 7 is a flowchart illustrating a method of determining pole pairnumbers for two magnetic rings when the outer radial boundary islimiting;

FIG. 8 is a graph illustrating theoretical and actual combined positionsensor readings; and

FIG. 9 illustrates arrangements for mounting the disc;

FIG. 10 illustrates the magnetic field detected by a magnetic sensorarray;

FIG. 11 illustrates theoretical and actual sensor readings taken whilsta single magnetic pole pair passes a magnetic sensor assembly;

FIG. 12 illustrates a correcting function for a pole pair; and

FIG. 13 illustrates theoretical and actual sensor readings taken whilsta whole magnetic ring passes a magnetic sensor assembly.

DETAILED DESCRIPTION

The following relates to a position sensing arrangement for anarticulated structure, and a method of assembling the position sensingarrangement. By sensing the position of each joint of an articulatedstructure, the position of the distal end of the articulated structurecan be determined from a combination of the sensed joint positions andknown layout of the articulated structure. In the example of a robotarm, a base of the robot arm is coupled to the end effector at thedistal end of the robot arm via a series of links joined together byjoints. These may be revolute joints or prismatic joints. In the case ofa revolute joint, the rotation of the joint is sensed. In other words,the relative rotation of the two shafts which the revolute jointattaches is sensed. The angle of rotation and the direction of rotationare sensed. In the case of a prismatic joint, the longitudinal motion ofthe joint is sensed. In other words, the relative motion of the twoshafts which the prismatic joint attaches is sensed. The movementdistance and the direction of movement are sensed.

FIG. 1 illustrates an example of a position sensing arrangement fordetecting rotation of shaft 1 about axis 2. The position sensingarrangement detects the angle and direction of rotation of shaft 1 aboutaxis 2. The position sensing arrangement comprises a magnetic sensorassembly and a disc 3. Disc 3 is shown in more detail on FIG. 2. Disc 3is an annulus having an outer radial boundary 4 and an inner radialboundary 5. Both the boundaries of the annulus are centred on the centrepoint of the disc 3. The radius of the outer radial boundary is r_(o).The radius of the inner radial boundary is r_(i). Disc 3 is fast withthe element whose position is being sensed. In this case, disc 3 isrigidly mounted to shaft 1. There is no relative motion permittedbetween shaft 1 and disc 3. Disc 3 rotates about axis 2. In other words,disc 3 and shaft 1 rotate about a common axis.

Two magnetic rings 6,7 are disposed on disc 3. The magnetic rings arenot movable relative to each other. The two magnetic rings areconcentric. Both magnetic rings are centred on the centre of the disc.In other words, the magnetic rings are arranged in a circle having therotation axis 2 of shaft 1 as its axis. The radial distance between thecentre of the disc and the centreline 9 of the inner magnetic ring 6 isr_(m). The radial distance between the centre of the disc and thecentreline 10 of the outer magnetic ring 7 is r_(n). The centrelines 9,10 of the magnetic rings 6,7 are separated by a radial distance s. Theminimum value of the radial distance s is predetermined. Suitably,radial distance s is at least the length of a pole pair. In other words,s≥2y.

Each magnetic ring carries a number of permanent magnets definingmagnetic poles 8. On the sensing surface of each magnetic ring, themagnets alternate polarity between north and south poles around thering. Inner magnetic ring 6 has m magnetic pole pairs. Outer magneticring 7 has n magnetic pole pairs. Each magnetic pole 8 on the innermagnetic ring 6 is the same shape and size, within manufacturingtolerance. Each magnetic pole 8 on the outer magnetic ring 7 is the sameshape and size, within manufacturing tolerance. Suitably, each magneticpole 8 on the inner magnetic ring 6 is the same shape and size as eachmagnetic pole 8 on the outer magnetic ring 7, within manufacturingtolerance and the fact that the arc radius of the inner magnetic ring isdifferent to the arc radius of the outer magnetic ring. A portion of amagnetic ring is shown in more detail in FIG. 3. Each magnetic pole 8has a radial length x and a circumferential length y. In one example, yis 2 mm. In this example, a pole pair (i.e. a north pole and adjacentsouth pole pair) has a circumferential length of 4 mm.

The magnetic sensor assembly is mounted to the articulated structure soas to detect relative rotation between two elements. The magnetic sensorassembly is rigidly attached to one of those elements, such that themagnetic sensor assembly is not permitted to move relative to thatelement. The disc 3 is rigidly attached to the other of those elements,such that the magnetic rings 6, 7 are not permitted to move relative tothat other element. In the case of the example of FIG. 1, the disc 3 isrigidly attached to the shaft 1. The magnetic sensor assembly is rigidlyattached to the part of the articulated structure that the shaft 1rotates relative to.

The magnetic sensor assembly detects relative rotation of the first andsecond magnetic rings and the magnetic sensor assembly. Magnetic sensorassembly comprises two magnetic sensor arrays 11,12. Inner magneticsensor array 11 is disposed adjacent to and aligned with the innermagnetic ring 6. Outer magnetic sensor array 12 is disposed adjacent toand aligned with the outer magnetic ring 7. Since the magnetic sensorassembly is mounted to the articulated structure relative to which shaft1 rotates, as shaft 1 rotates, the magnetic rings 6 and 7 revolve pastthe magnetic sensor arrays 11, 12. Each sensor array is capable ofdetecting transitions between north and south poles of the magnetic ringit is disposed over as those transitions move past the sensor array. Inan exemplary implementation, the first and second magnetic rings 6, 7are separated radially by at least the length of a pole pair. Increasingthe separation of the rings reduces the interference each ring causes tothe sensor of the other ring. Thus, separating the rings by at least thelength of a pole pair aids the inner magnetic sensor array 11 onlydetecting the transitions of the inner magnetic ring 6, and the outermagnetic sensor array 12 only detecting the transitions of the outermagnetic ring 7.

Each magnetic sensor array 11, 12 comprises a set of sensors. FIG. 4illustrates an example in which a magnetic sensor array comprises fourindividual sensors 13 a,b,c,d. Each magnetic sensor array isrectilinear. As can be seen from FIG. 4, the individual sensors arearranged in a straight line. The sensors of the set are all the samesize and shape and have equal spacing between them. Each sensor 13 has awidth t and a length u and is separated from the next sensor by adistance v. The centres of adjacent sensors are separated by a distancez. In the example shown in FIG. 4 in which the magnetic sensor array hasfour individual sensors, z is the same as a quarter the length of a polepair. In other words, z=y/2. Thus, whilst the centres of the sensorsmarked 1 and 3 are over the boundary between adjacent poles, the centresof sensors marked 2 and 4 are over the centres of adjacent poles. Thecentres of the outer sensors (marked 1 and 4) are separated by threequarters the length of a pole pair. In an exemplary implementation, t isless than the radial extent of the magnetic ring x. In other words, t<x.

Since the individual sensors are in a straight line whereas the magneticring is circular, the centre of each sensor is not consistently alignedwith the centre of a magnetic pole as the shaft rotates. The offsetbetween the centre of the sensor and the centre of the magnetic polevaries as the shaft rotates. This variable offset causes a systematicerror in the sensor output. In an alternative implementation, themagnetic sensor arrays 11, 12 are each in a circular configurationcentred on the centre of the disc 3. In this case, the radius of thecentreline of each magnetic sensor array is the same as the radius ofthe centreline of the magnetic ring it is reading. Thus, the centrelineof the magnetic sensor array is consistently aligned with the centrelineof the magnetic ring it is reading as the shaft rotates.

The sensors could, for example, be Hall effect sensors, reed sensors,magnetoresistive sensors or inductive sensors.

Each magnetic sensor array 11, 12 is arranged to provide a multi-bitoutput representing the relative position of the neighbouring poles toit. The number and relative placement of the poles on the magnetic ringsare arranged such that each position of the shaft within the range ofrotation angles to be measured is associated with a unique set ofoutputs from the two magnetic sensor arrays 11, 12. The number of polesm on the inner ring and the number of poles n on the outer ring aredifferent and co-prime. Their selection is described further below. Theoutputs from the sensors pass to a processing unit 14.

The circumferential positions of the magnetic sensor arrays 11, 12 andthe rotational position of the disc 3 about axis 2 may be chosen so thatthe transitions between the poles on the inner magnetic ring 6 as sensedby magnetic sensor array 11 occur for different rotational positions ofthe shaft from the transitions between the poles on the outer magneticring 7 as sensed by magnetic sensor array 12. This allows the directionof rotation of the shaft to be inferred from the relative order of thetransitions sensed by each magnetic sensor array.

The outputs of magnetic sensor arrays 11, 12 pass to the processing unit14. The processing unit comprises a processor device 15, which could behard coded to interpret the signals from the magnetic sensor arrays 11,12 or could be a general purpose processor configured to executesoftware code stored in a non-transient way in memory 16. The processordevice combines the signals from the sensors to form an integratedoutput signal at 17.

A method of selecting the number of magnetic pole pairs m on the innermagnetic ring 6 and the number of magnetic pole pairs n on the outermagnetic ring 7 will now be described.

The selection of m and n may be subject to any one, any combination, orall of the following constraints.

-   -   1. The inner and outer magnetic rings 6, 7 both need to fit on        the disc 3. Suitably, the inner and outer magnetic sensor arrays        11, 12 are each disposed within the footprint of the disc 3 as        well. The inner radial boundary 5 is at a radius r_(i). In order        to fit the inner magnetic sensor array 11 over the disc 3        without exceeding the inner radial boundary 5, the centreline 9        of the inner magnetic ring is separated from the inner radial        boundary 5 by at least the portion of the radial width of the        magnetic sensor assembly 11 which is disposed between the        centreline 9 of the inner magnetic ring and the inner radial        boundary 5. This portion may be half the radial width of the        magnetic sensor assembly, i.e. w_(m)/2. In other words,        r_(m)>r_(i)+w_(m)/2. If the sensor array is offset radially        within the sensor assembly, then the portion may be greater. In        one example, r_(m)>r_(i)+(w_(m)+t)/2. The outer radial boundary        4 is at a radius r_(o). In order to fit the outer magnetic        sensor array 12 over the disc 3 without exceeding the outer        radial boundary 4, the centreline 10 of the outer magnetic ring        is separated from the outer radial boundary 4 by at least the        portion of the radial width of the magnetic sensor assembly 12        which is disposed between the centreline 10 of the outer        magnetic ring and the outer radial boundary 4. This portion may        be the radial width of the magnetic sensor assembly, i.e.        w_(n)/2. In other words, r_(n)<r_(n)−w_(n)/2. If the sensor        array is offset radially within the sensor assembly, then the        portion may be greater. In one example, r_(n)<r_(o)−(w_(n)+t)/2.    -   2. The centres of adjacent sensors are separated by a quarter        the length of a pole pair. In other words, adjacent sensors are        separated by y/2. This ensures that the magnetic sensor array        can detect each pole transition as the disc rotates.    -   3. The radial distance s between the centreline of the inner        magnetic ring 6 and the centreline of the outer magnetic ring 7        is greater than a predetermined distance. That predetermined        distance is set by a desired insensitivity to stray magnetic        fields. Suitably, s>2y. In this case, the nearest magnetic pole        on the inner magnetic ring 6 that the inner magnetic sensor        array 11 is detecting is always closer than a magnetic pole on        the outer magnetic ring 7. Similarly, the nearest magnetic pole        on the outer magnetic ring 7 that the outer magnetic sensor        array 12 is detecting is always closer than a magnetic pole on        the inner magnetic ring 6. Thus, this prevents interference        arising from the magnetic sensor array over one magnetic ring        detecting a magnetic field from the other magnetic ring.    -   4. The minimum number of sensors b in each magnetic sensor array        is greater than a threshold. This threshold is such that there        is sufficient spatial sampling for an unambiguous position        reading. Suitably, b≥4. Four sensors is sufficient where there        are no magnetic harmonics detected.    -   5. The width, t, of each magnetic sensor is less than the radial        extent of the magnetic ring x. In other words, t<x. The signal        to noise ratio of the detected transitions is reduced if the        magnetic sensor array is narrower than the magnetic ring.    -   6. The inner radial boundary of the disc may be constrained by        the articulated structure. For example, in FIG. 1, the inner        radial boundary of the disc is limited by the shaft 1. In this        case, the radius of the inner radial boundary r_(i) has to be at        least as big as the radius of the shaft 1.    -   7. The outer radial boundary of the disc may be constrained by        the articulated structure. For example, the position sensor may        fit inside a housing of the articulated structure. In this case,        the radius of the outer radial boundary r_(o) has to be at least        as small as the radius of the housing.    -   8. The position sensor is arranged to detect a maximum angle of        rotation. This maximum angle of rotation depends on the element        whose rotation is being detected. For a revolute joint, the        maximum angle of rotation to be detected depends on the location        of that joint in the kinematic chain. The maximum angle of        rotation to be detected may be less than 360°. The maximum angle        of rotation to be detected may be greater than 360°. The        accuracy of the position measurement is proportional to the        angle of rotation to be detected. The greater the angle of        rotation to be detected, the higher the accuracy of the position        measurement needed. The sensor reading accuracy is given by:

Accuracy=±2y×1/2×1/Σpole pairs   (equation 1)

Σ pole pairs is the sum of the number of pole pairs of all of themagnetic rings on the disc. When there are two magnetic rings on thedisc, Σ pole pairs=m+n. The more pole pairs on the magnetic rings, thelarger the magnetic rings. Thus, larger magnetic rings lead to moreaccurate position measurements. The greater the angle of rotation to bedetected, the bigger m and/or n are to achieve the required accuracy.Thus, the number of pole pairs on the magnetic rings for detectingrotation of one element relative to another is constrained by therelative angle of rotation to be detected between the two elements. Inthe case of a revolute joint, the selection of m and n are specific tothe maximum angle of rotation of that revolute joint to be detected.

-   -   FIG. 5 is a graph which illustrates theoretical and actual        position sensor measurements taken from a position sensing        arrangement of the form shown in FIG. 1. The x-axis is the        position sensor measurement of the outer magnetic sensor array        12, and the y-axis is the position sensor measurement of the        inner magnetic sensor array 11. The starred line plot        illustrates theoretical measurements with 100% accuracy. The        solid line plot illustrates example actual readings. These        example actual readings differ from the theoretical readings due        to manufacturing variation in the magnetisation of the magnetic        rings on the disc, manufacturing variation in the magnetic        sensor arrays, and/or misalignment between the magnetic rings        and the magnetic sensor arrays. In FIG. 5 the actual readings        have a consistent offset from the theoretical readings which        demonstrates a systematic error in the actual readings. This        offset may be due to an error when magnetising the disc. For        example, the centre of the magnetic rings may be offset slightly        from the centre of rotation of the disc. FIG. 5 also        demonstrates additional error in the readings beyond the        systematic error. In order to accurately detect which sensor        reading was intended, the solid line of the actual readings in        the plot of FIG. 5 needs to be closer to the correct line of        theoretical readings than another line of theoretical readings.

A method of determining the values of m and n will now be described foran example in which the inner radial boundary of the disc 3 is limiting.For example, the disc may be mounted on a shaft, and hence the radius ofthe inner radial boundary r_(i) of the disc 3 has to be greater than theradius of the shaft. FIG. 6 illustrates the steps of this method.

At step 20, the minimum r_(m,min) is determined. This is the minimumradius of the centreline 9 of the inner magnetic ring permitted by thelimiting inner radius r_(i). As described in constraint 1 above, at itsminimum, the centreline 9 of the inner magnetic ring is separated fromthe inner radial boundary 5 by the portion of the radial width of themagnetic sensor assembly 11 which is disposed between the centreline 9of the inner magnetic ring and the inner radial boundary 5. This is toensure that the magnetic sensor assembly 11 is confined within the innerradial boundary. In one example, the minimum r_(m-min) is given by:

r _(m,min) =r _(i)+(w _(m) +t)/2   (equation 2)

At step 21, the minimum number of pole pairs m for the inner magneticring is determined. The magnetic ring has a whole number of pole pairs.Thus, m is an integer. r_(m,min) is increased to the lowest value ofr_(m), where

2πr_(m)=m2y   (equation 3)

Where m is an integer.

At step 22, the minimum r_(n,min) is determined. This is the minimumradius of the centreline 10 of the outer magnetic ring permitted by thelimiting inner radius r_(i). As described in constraint 3 above, at itsminimum, the centreline 10 of the outer magnetic ring is separatedradially from the centreline 9 of the inner magnetic ring by apredetermined distance. This predetermined distance is suitably largeenough to reduce or minimise interference in the sensor reading of onemagnetic ring as a result of the other magnetic ring.

r_(n,min) =r _(m) +s   (equation 4)

where r_(m) is that from equation 3, and s is the predetermineddistance.

At step 23, the minimum number of pole pairs n for the outer magneticring is determined. The magnetic ring has a whole number of pole pairs.Thus, n is an integer. r_(n,min) is increased to the lowest value ofr_(n), where

2πr_(n)=n2y   (equation 5)

Where n is an integer.

At step 24, it is determined whether the values of m and n determined atsteps 21 and 23 are co-prime. If m and n are co-prime, then these arethe m,n pair which provide the most compact disc. In this case, m ischosen to be the number of pole pairs on the inner magnetic ring, and nis chosen to be the number of pole pairs on the outer magnetic ring. Themethod proceeds to step 25 where the position sensing arrangement isconstructed by mounting a disc having an inner magnetic ring with m polepairs and an outer magnetic ring with n pole pairs to the articulatedstructure. The disc is rigidly attached to the element of thearticulated structure whose position it is configured to sense. The discis mounted such that it rotates about the same axis as the element whoseposition it is configured to sense.

If, at step 24, it is determined that the values of m and n determinedat steps 21 and 23 are not co-prime, then the method proceeds to step26. At step 26, the value of n determined at step 23 is incrementedby 1. At step 27, it is determined whether the new value of n determinedat step 26 and the value of m determined at step 21 are co-prime. Ifthey are not co-prime, then the method returns to step 26, where thevalue of n is incremented by 1. Then the method returns to step 27 whereit is determined whether the new value of n is co-prime with m. Steps 26and 27 continue iteratively, each iteration incrementing the value of nby 1, until a value of n is reached which is co-prime with m. Eachiteration thereby increments the difference between m and n by 1.

Once a value of n is found which is co-prime with m, the method proceedsto step 28. At step 28, the value of m determined at step 21 is set asthe lower bound of a range of number of pole pairs of the inner magneticring 6. Also at step 28, the value of n which was determined to beco-prime with m at step 27 is set as the upper bound of a range ofnumber of pole pairs of the outer magnetic ring 7.

At step 29, further co-prime m,n pairs are identified which lie withinthe ranges set by step 28. In other words, further values of m and n areidentified which are co-prime and for which m is greater than the lowerbound set in step 28 and n is smaller than the upper bound set in step28. These co-prime m,n pairs also satisfy any other constraints, such asbeing such that the outer magnetic ring is separated from the innermagnetic ring by at least the predetermined minimum value of thedistance s.

In an illustrative example, the m and n values initially determined atsteps 21 and 23 are m=30 and n=35. These are not co-prime. Iterating nat step 26 results in an m,n pair of m=30 and n=37. At step 29 thefollowing further co-prime pairs are identified: m=31, n=36; m=31, n=37;m=32, n=37.

At step 30, one of the co-prime m,n pairs is selected. This m,n pair maybe either the value of m determined in step 21 and the value of ndetermined to be co-prime with that value of m in step 27, oralternatively, the m,n pair may have been determined at step 29. The m,npair selected depends on the implementation.

In one example, the co-prime m,n pair having the largest value of m isselected at step 30. In the example provided above, m=32, n=37 would beselected. Selecting the m,n pair with the largest value of m maximisesthe accuracy of the resulting sensor, as can be seen from equation 1.

In another example, the co-prime m,n pair having the smallest value of nis selected at step 30. In the example provided above, m=31, n=36 wouldbe selected. Selecting the m,n pair with the smallest value of nminimises the outer radius of the disc r_(o), thus minimises the overallspace taken up by the sensor.

In another example, the co-prime m,n pair having the smallest value ofn-m is selected at step 30. Selecting the m,n pair with the smallestvalue of n-m minimises the radial width of the disc, thus provides themost compact sensor. In the example provided above, two m,n pairs havethe smallest value of n-m: m=31, n=36; and m=32, n=37.

The m,n pair selected at step 30 may be chosen in dependence on themaximum angle of rotation to be sensed. As described above, the maximumangle of rotation specifies a required minimum accuracy, which in turnspecifies a required minimum sum of the number of pole pairs on themagnetic rings. This is a competing constraint to providing a compactsensor. To balance these competing requirements, the smallest co-primem,n pair whose sum exceeds the required minimum sum of the number ofpole pairs on the magnetic rings may be selected at step 30.

Any one or combination of these criteria may be applied in a specificimplementation. For example, the co-prime m,n pair having the smallestvalue of n-m may be selected. In the event that there is more than oneco-prime m,n pair having the smallest value of n-m, the one of thosewhich has the largest m may be selected. Thus, in the example above,m=32, n=37 would be chosen.

Once the co-prime m,n pair are selected at step 30, m of the selectedm,n pair is chosen to be the number of pole pairs on the inner magneticring, and n of the selected m,n pair is chosen to be the number of polepairs on the outer magnetic ring. The method then proceeds to step 25,where the position sensing arrangement is assembled by mounting a dischaving an inner magnetic ring with m pole pairs and an outer magneticring with n pole pairs to the articulated structure as previouslydescribed.

A method of determining the values of m and n will now be described foran example in which the outer radial boundary of the disc 3 is limiting.For example, the disc may be mounted within a housing, and hence theradius of the outer radial boundary r_(o) of the disc 3 has to be lessthan the closest distance of the housing to the centre point of thedisc. FIG. 7 illustrates the steps of this method.

At step 40, the maximum r_(n,max) is determined. This is the maximumradius of the centreline 10 of the outer magnetic ring permitted by thelimiting outer radius r_(o). As described in constraint 1 above, at itsmaximum, the centreline 10 of the outer magnetic ring is separated fromthe outer radial boundary 4 by the portion of the radial width of themagnetic sensor assembly 12 which is disposed between the centreline 10of the outer magnetic ring and the outer radial boundary 4. This is toensure that the magnetic sensor assembly 12 is confined within the outerradial boundary. In one example, the maximum r_(n,max) is given by:

r _(n,max) =r _(o)−(w _(n) +t)/2   (equation 6)

At step 41, the maximum number of pole pairs n for the outer magneticring is determined. The magnetic ring has a whole number of pole pairs.Thus, n is an integer. r_(n,max) is decreased to the highest value ofr_(n), where

2πr_(n)=n2y   (equation 7)

Where n is an integer.

At step 42, the maximum r_(m,max) is determined. This is the maximumradius of the centreline 9 of the inner magnetic ring permitted by thelimiting outer radius r_(o). As described in constraint 3 above, at itsminimum, the centreline 9 of the inner magnetic ring is separatedradially from the centreline 10 of the outer magnetic ring by apredetermined distance. This predetermined distance is suitably largeenough to reduce or minimise interference in the sensor reading of onemagnetic ring as a result of the other magnetic ring.

r _(m,max) =r _(n) −s   (equation 8)

where r_(n) is that from equation 7, and s is the predetermineddistance.

At step 43, the maximum number of pole pairs m for the inner magneticring is determined. The magnetic ring has a whole number of pole pairs.Thus, m is an integer. r_(m,max) is decreased to the largest value ofr_(m), where

2πr_(m)=m2y   (equation 9)

Where m is an integer.

At step 44, it is determined whether the values of m and n determined atsteps 41 and 43 are co-prime. If m and n are co-prime, then these arethe m,n pair which provide the most compact disc. In this case, m ischosen to be the number of pole pairs on the inner magnetic ring, and nis chosen to be the number of pole pairs on the outer magnetic ring. Themethod proceeds to step 45 where the position sensing arrangement isconstructed by mounting a disc having an inner magnetic ring with m polepairs and an outer magnetic ring with n pole pairs to the articulatedstructure. The disc is rigidly attached to the element of thearticulated structure whose position it is configured to sense. The discis mounted such that it rotates about the same axis as the element whoseposition it is configured to sense.

If, at step 44, it is determined that the values of m and n determinedat steps 41 and 43 are not co-prime, then the method proceeds to step46. At step 46, the value of m determined at step 43 is decrementedby 1. At step 47, it is determined whether the new value of m determinedat step 46 and the value of n determined at step 41 are co-prime. Ifthey are not co-prime, then the method returns to step 46, where thevalue of m is decremented by 1. Then the method returns to step 47 whereit is determined whether the new value of m is co-prime with n. Steps 46and 47 continue iteratively, each iteration decrementing the value of mby 1, until a value of m is reached which is co-prime with n. Eachiteration thereby increments the difference between m and n by 1.

Once a value of m is found which is co-prime with n, the method proceedsto step 48. At step 48, the value of n determined at step 41 is set asthe upper bound of a range of number of pole pairs of the outer magneticring 7. Also at step 48, the value of m which was determined to beco-prime with n at step 47 is set as the lower bound of a range ofnumber of pole pairs of the inner magnetic ring 6.

At step 49, further co-prime m,n pairs are identified which lie withinthe ranges set by step 48. In other words, further values of m and n areidentified which are co-prime and for which m is greater than the lowerbound set in step 48 and n is smaller than the upper bound set in step48. These co-prime m,n pairs also satisfy any other constraints, such asbeing such that the outer magnetic ring is separated from the innermagnetic ring by at least the predetermined distance.

At step 50, one of the co-prime m,n pairs is selected. This m,n pair maybe either the value of n determined in step 41 and the value of mdetermined to be co-prime with that value of m in step 47, oralternatively, the m,n pair may have been determined at step 49. The m,npair selected depends on the implementation.

In one example, the co-prime m,n pair having the largest value of m isselected at step 40. Selecting the m,n pair with the largest value of mmaximises the accuracy of the resulting sensor, which can be seen fromequation 1.

In another example, the co-prime m,n pair having the smallest value ofn-m is selected at step 50. Selecting the m,n pair with the smallestvalue of n-m minimises the radial width of the disc, thus provides themost compact sensor.

The m,n pair selected at step 50 may be chosen in dependence on themaximum angle of rotation to be sensed. The smallest co-prime m,n pairwhose sum exceeds the required minimum sum of the number of pole pairson the magnetic rings may be selected at step 50.

Any one or combination of these criteria may be applied in a specificimplementation. For example, the co-prime m,n pair having the smallestvalue of n-m may be selected. In the event that there is more than oneco-prime m,n pair having the smallest value of n-m, the one of thosewhich has the largest m may be selected.

Once the co-prime m,n pair are selected at step 50, m of the selectedm,n pair is chosen to be the number of pole pairs on the inner magneticring, and n of the selected m,n pair is chosen to be the number of polepairs on the outer magnetic ring. The method then proceeds to step 45,where the position sensing arrangement is assembled by mounting a dischaving an inner magnetic ring with m pole pairs and an outer magneticring with n pole pairs to the articulated structure as previouslydescribed.

It will be understood that the flowcharts of FIGS. 6 and 7 representexemplary methods of determining the number of poles on the inner andouter magnetic rings in cases for which the inner radial boundary or theouter radial boundary of the disc is limiting. Not all of the stepsdescribed are necessarily required to determine the number of poles. Forexample, the minimum number of poles m and n determined at steps 21 and23 of FIG. 6 could be determined without actually determining theminimum radii r_(m,min) and r_(n,min). Similarly, the maximum number ofpoles m and n determined at steps 41 and 43 of FIG. 7 could bedetermined without actually determining the maximum radii r_(m,max) andr_(n,max).

The angle of rotation detected by the sensor can be determined from thesensor readings from the inner magnetic ring and the outer magnetic ringas follows. The angle of rotation detected is equal to the number ofwhole revolutions of the outer magnetic ring plus the current sensorreading for the outer magnetic ring. FIG. 8 is a graph which illustratestheoretical and actual sensor measurements taken from a position sensingarrangement of the form shown in FIG. 1. The x-axis is the positionsensor measurement of the outer magnetic sensor array 12. The y-axis isa combined sensor reading which is ((outer sensor array reading xn)−(inner sensor array reading×m)). The starred line plot illustratestheoretical measurement with 100% accuracy. The solid line plotillustrates example actual readings. The starred line plot is a seriesof straight lines. Each straight line represents a specific number ofrevolutions of the outer magnetic ring and a specific number ofrevolutions of the inner magnetic ring. Thus, one method of determiningthe angle of rotation detected by the sensor is to determine:

X=(outer sensor array reading×n)−(inner sensor array reading×m)  (equation 10)

And compare X to a look up table which maps X to a number of revolutionsof the outer ring. The angle of rotation is then:

Angle of rotation=no. of revolutions of outer ring +current outer sensorarray reading   (equation 11)

The angle of rotation could alternatively be determined in a similarmanner with respect to the inner sensor array readings.

In the example shown in FIGS. 1 and 2, both magnetic rings 6 and 7 aredisposed on the same surface of disc 3. However, magnetic rings 6 and 7may be disposed on opposing surfaces of disc 3. In this case, themagnetic sensor arrays are mounted over opposing surfaces of the disc 3.The inner magnetic sensor array is mounted over the inner magnetic ringso as to detect transitions of poles on that inner magnetic ring. Theouter magnetic sensor array is mounted over the outer magnetic sensorarray so as to detect transitions of poles on that outer magnetic ring.In this case, the minimum radial spacing s between the centrelines 9 and10 of the inner and outer magnetic rings is less than if the magneticrings are on the same surface of the disc. This is because theinterference caused to a magnetic sensor array by a magnetic ring whosepoles are exposed on the opposing surface of the disc to the magneticsensor array is less than the interference caused by a magnetic ring thesame distance away on the same side of the disc as the magnetic sensorarray. Thus the radial distance between the centrelines of the inner andouter magnetic rings may be less than the length of a pole pair 2y.

In the example shown in FIGS. 1 and 2, the inner and outer magneticrings 6,7 are on the same disc 3. Alternatively, the inner magnetic ringand the outer magnetic ring may be disposed on different discs. Thosedifferent discs have the same axis of rotation, which is the same as theelement whose rotation is to be detected. However, those different discsare separated along the axis of rotation. For example, when measuringthe rotation of a revolute joint, the discs may be mounted on opposingsides of the joint. This may be desirable for packing/compactnessreasons. In this example, the number of pole pairs m and non the innerand outer magnetic rings would be determined as described herein.However, since the inner and outer magnetic rings are axially separated,the interference caused by one magnetic ring to the sensing of the othermagnetic ring is negligible, and hence there is no constraint for themagnetic rings to be separated radially.

The apparatus and methods described herein can be used to detect lessthan a full revolution of one element relative to another. The apparatusand methods described herein can also be used to detect greater than onefull revolution of one element with respect to another.

In an example implementation, a motor drives a gearbox which drives anelement of the articulated structure. One magnetic ring is rigidlyattached to the motorshaft output from the motor. The other magneticring is rigidly attached to the driveshaft output from the gearbox.Thus, the two magnetic rings are axially separated. The axis of rotationof the motorshaft and the driveshaft are the same. Both magnetic ringsare centred on this axis of rotation. Suitably, the inner magnetic ringhaving m poles is attached to the motorshaft, and the outer magneticring having n poles is attached to the driveshaft. n>m. If a fractionalgear ratio is used, then more than one revolution of the driveshaft canbe distinguished. For example, with a gear ratio of 13:4, up to fourrevolutions of the driveshaft can be distinguished. This is useful in arobot implementation because it enables the position of the driveshaftto be determined during setup without having to fully rotate thedriveshaft one direction and then the other.

In an arrangement in which m is less than a threshold, the magneticsensor array 11 for the inner magnetic ring is implemented as an on-axiscircular array rather than a linear array. The threshold is the maximumnumber of poles m for which the linear magnetic sensor array does notintersect the arc of the centreline of the inner magnetic ringsufficiently to provide a useable reading.

In one implementation, m=1. In this case, there is just one pole pair onthe inner magnetic ring. This pole pair is centred on the centre of thedisc 3. This pole pair is on the rotation axis of the disc. This polepair is not aligned with any of the pole pairs on the outer magneticring. This enables the direction of rotation to be detected. Using m=1enables a very compact sensor to be used, however it does have lowaccuracy (see equation 1).

As described above, the m,n pair selected are co-prime. In all casesn−m>1. In the case that the inner and outer magnetic rings 6,7 are onthe same surface of disc 3, n−m≥7. This is because the radial distance sbetween the centrelines of the inner and outer magnetic rings is greaterthan or the same as the length of a pole pair to avoid interference. Thenumber of pole pairs on each magnetic ring is an integer. Thus, theminimum difference between the number of pole pairs on the two magneticrings is the first integer greater than 2π, which is 7. In one example,n−m≥8. For example, n=41, m=33. In another example, n−m≥10. For example,n=47, m=37.

The apparatus and methods described above relate to sensing the rotationof two magnetic rings 6, 7. The same methods can be adapted to sensingthe rotation of three or more magnetic rings. All the magnetic rings areco-axial. They may all be disposed on the same disc or several discsseparated axially along their rotation axis. Equation 1 shows that theaccuracy of the position sensed increases as the sum of the pole pairson the magnetic rings increases. Thus by utilising further magneticrings, a more accurate position is measured. The number of pole pairs oneach magnetic ring is co-prime with the number of pole pairs on all theother magnetic rings. Thus, in the case that there are three magneticrings, having m, n and h pole pairs, m, n and h are all co-prime. Eachmagnetic ring is suitably radially separated from adjacent magneticrings by at least the predetermined distance. Where the magnetic ringsare all on the same surface of a disc, each magnetic ring is preferablyseparated from adjacent magnetic rings by at least the length of a polepair 2y.

In the case of FIG. 6, for a position sensor having more than twomagnetic rings, a minimum m and a minimum n would be determined as laidout in steps 21 and 23. If there is a third magnetic ring, then aminimum h would be determined in the same manner as n was determinedthrough steps 22 and 23 but this time the minimum radial separation isbetween the ring with n pole pairs and the ring with h pole pairs.Similarly, if there are further magnetic rings, steps 22 and 23 arerepeated for each further magnetic ring in a corresponding manner. Onlyif all the pole pair numbers are co-prime at step 24, does the methodproceed to step 25. Otherwise, method steps 26 and 27 are performed foreach further ring until a set of co-prime pole pair numbers for eachring is determined. The pole pair number of the smallest magnetic ringis set as the low bound for that magnetic ring at step 28. The pole pairnumber for the largest magnetic ring is set as the high bound for thatmagnetic ring at step 28. Further co-prime pole pair numbers aredetermined for all the magnetic rings at step 29, ensuring that theseparation between the adjacent rings is at least the minimum radialseparation. A pole pair combination is selected at step 30 in accordancewith the above-described methods. The position sensing arrangement isconstructed by mounting a disc to the articulated structure which has agroup of magnetic rings with the numbers of pole pairs on each ringselected at step 30.

In the case of FIG. 7, for a position sensor having more than twomagnetic rings, a maximum n and a maximum m would be determined as laidout in steps 41 and 43. If there is a third magnetic ring, then amaximum h would be determined in the same manner as m was determinedthrough steps 42 and 43 but this time the minimum radial separation isbetween the ring with m pole pairs and the ring with h pole pairs.Similarly, if there are further magnetic rings, steps 42 and 43 arerepeated for each further magnetic ring in a corresponding manner. Onlyif all the pole pair numbers are co-prime at step 44, does the methodproceed to step 45. Otherwise, method steps 46 and 47 are performed foreach further ring until a set of co-prime pole pair numbers for eachring is determined. The pole pair number of the smallest magnetic ringis set as the low bound for that magnetic ring at step 48. The pole pairnumber for the largest magnetic ring is set as the high bound for thatmagnetic ring at step 48. Further co-prime pole pair numbers aredetermined for all the magnetic rings at step 49, ensuring that theseparation between the adjacent rings is at least the minimum radialseparation s. A pole pair combination is selected at step 50 inaccordance with the above-described methods. The position sensingarrangement is assembled by mounting a disc to the articulated structurewhich has a group of magnetic rings with the numbers of pole pairs oneach ring selected at step 50.

The position sensor described herein is capable of absolutedetermination of the relative rotational position of two objects. Inother words, the position can be determined directly from the output ofthe sensors without, for example, the need to count up the motion sincethe relative rotational position was in a reference configuration.

The apparatus and methods described with reference to FIGS. 1 and 2relate to measuring the relative rotation of two elements. However, thesame principles apply to measuring relative linear motion of twoelements. In this case, rather than the magnetic array being arranged asrings of magnetic pole pairs, they are arranged as linear tracks ofmagnetic pole pairs. Two or more linear tracks are used. The tracks haveco-prime numbers of magnetic pole pairs. Each track lies parallel to thelinear motion of the element whose position is being sensed. Arespective linear magnetic sensor array is mounted over each magnetictrack, such that as the element moves, the linear magnetic sensor arraydetects the pole transitions of that linear magnetic sensor array. Forexample, for an element which moves linearly within a housing, themagnetic sensor arrays may be mounted to the element, and the magnetictracks fixed to the housing.

At manufacture, disc 3 is magnetised to cause the magnetic rings 6, 7 tohave the layout described above. Magnetisation heads are mounted overthe disc in the same manner as the magnetic sensor arrays 11, 12 of FIG.1 to magnetise the disc. A magnetic back plate is positioned on theopposing surface of the disc to the surface that the magnetisation headsare mounted over. The disc is rotated, thereby magnetising each polepair of the magnetic rings.

The accuracy with which the poles are positioned on the disc is limitedby manufacturing errors. Those errors include radial positioning errorsand angular spacing errors. Radial positioning errors occur when thereis an offset between the centre of the magnetic rings and the rotationaxis of the disc when it is mounted to the articulated structure. Whenmounted, the centreline of each magnetic ring will not be at a constantradius from the rotation axis. When used in a position sensor, theradius of the centreline of each magnetic ring from the rotation axis isvariable around its circumference, and hence different for differentpole pairs of the magnetic ring. Radial positioning errors also occur ifthere is an offset between the intended and actual radial positions ofthe magnetisation head. In such a case, the magnetic ring induced by themagnetisation head has a constant radius from the centre of the disc,but not the intended radius. Thus, a whole number of pole pairs does notfit around the circumference of the magnetic ring, which leads to unevenlengths of one or more pole pairs of the magnetic ring.

The pole pairs of each magnetic ring are intended to have a constantcircumferential length of 2y. Angular spacing errors occur when thecircumferential length of the poles are not even around the magneticring. This may occur if the disc was not rotated evenly under themagnetisation heads during magnetisation. If the poles are not ofuniform length, then the sensed position will be inaccurate.

Radial positioning errors and angular spacing errors result in a polepattern that is irregular and not concentric to the rotation axis. Theaccuracy of the position measurement required depends on what it isbeing used for. In the field of robotics, particularly surgicalrobotics, the position measurements need to be highly accurate. Theposition measurements of all the joints of the robot arm are used incombination with the known layout of the robot arm to determine theposition of the end effector. The position of the end effector needs tobe known with high precision in order to control it to performprocedures where fine control is required, such as suturing tissue in apatient. The position measurement may be required to have an accuracy of±25 μm, where the accuracy is determined by equation 1. As previouslydiscussed, the accuracy required varies with the angle of rotation whichneeds to be detectable. The greater the angle of rotation which needs tobe detectable, the greater the accuracy required.

All of the magnetic rings of the disc 3 are magnetised at the same timeon the same magnetisation jig. The magnetisation jig has as manymagnetisation heads as there are magnetic rings to be magnetised on thedisc. Each magnetisation head induces the magnetic pole pair pattern ofone magnetic ring. In an exemplary implementation, the differencebetween the placement of the poles on the two magnetic rings is accurateto within ±2y×½×1/Σpole pairs. By magnetising all the magnetic ringsusing the same set up at the same time, any radial positioning error isconsistently applied across all the magnetic rings. Similarly, anyangular spacing error due to the disc not being rotated at a uniformrate, will be consistently applied across all the magnetic rings. Thesesystematic errors affect the pole pairs of all the magnetic ringsequally, and therefore the error introduced to the difference betweenthe placement of the poles on the two magnetic rings will be smallerthan the sum of the errors introduced into the two rings individually.Thus, the tolerance required in the magnetisation when both rings aremagnetised together to achieve the desired accuracy is significantly(almost half) the tolerance required if the rings are magnetisedindividually. The systematic errors may be detectable in the positionmeasurements and compensated for. For example, referring to FIG. 5, aportion of the error in the actual data of the solid line plot issystematic, which can be seen from the fact that the solid line isconsistently offset from the theoretical data line. This offset islikely due to a magnetisation error of the type described above. Thisconstant offset can be determined and removed, to produce a moreaccurate result.

The magnetisation heads of the magnetisation jig may be positioned onthe same radial line from the centre of rotation of the disc when it ismounted to the magnetisation jig. This ensures that any error introducedby uneven rotation of the disc during magnetisation is applied along thesame radial line to all the magnetic rings. The magnetic sensor arraysof the position sensing arrangement are positioned on the same radialline from the centre of rotation of the disc when it is mounted to thearticulated structure. The magnetic sensor arrays are located in thesame position and orientation relative to the disc that themagnetisation heads are located in during magnetisation.

The sensors in the magnetic sensor arrays may be monolythic. By formingthe sensors in the same process, any error is consistent amongst thesensors, and hence more easily identified as a systematic error when thesensor readings are being evaluated to generate a position measurement.

The values m, n, h etc for the number of pole pairs on the magneticrings of the disc may be chosen to be as large as possible within theother constraints. This increases the accuracy of the subsequentmeasurements and hence reduces the concentricity and positional errorsintroduced by the manufacturing process.

The disc is mounted to the articulated structure in the sameconfiguration that it is mounted to the magnetisation jig. In otherwords, the disc is mounted to the articulated structure in the sameposition and orientation that it is mounted to the magnetisation jig.Suitably, there is only one single orientation in which the disc ismountable to the magnetisation jig and articulated structure. The disccomprises a mounting arrangement which aids the user to mount the discto the articulated structure in the same configuration that it wasmounted to the magnetisation jig.

FIG. 9 illustrates an exemplary mounting arrangement. The mountingarrangement comprises a set of through-holes 60 a, 60 b, 60 c, 60 d.These through-holes form an asymmetrical pattern on the disc. Thethrough-holes enable the disc to be mounted to complimentary features onthe magnetisation jig and the articulated structure. Those complimentaryfeatures are arranged in the same asymmetrical pattern as theasymmetrical pattern of the through-holes on the disc. For example, thedisc may mount to pins which protrude from the magnetisationjig/articulated structure in the same arrangement as the pattern ofthrough-holes on the disc. In this example, the disc is secured to themagnetisation jig/articulated structure by a retaining mechanism whichretains the disc against the magnetisation jig/articulated structure.For example, the pins may be threaded pins, and a threaded nut may bethreaded onto the threaded pin to retain the disc to the magnetisationjig/articulated structure. In another example, the magnetisationjig/articulated structure may have threaded recesses in the sameasymmetrical pattern as the pattern of through-holes on the disc. Thedisc may be retained to the magnetisation jig/articulated structure byscrewing the disc to the magnetisation jig/articulated structure throughthe through-holes into the threaded recesses.

FIG. 9 illustrates through-holes as the mounting features of themounting arrangement. However, other types of mounting features would beappropriate, as long as complimentary features are provided on themagnetisation jig/articulated structure in the same asymmetricalpattern, and as long as the disc can be secured to the magnetisationjig/articulated structure by the mounting features and complimentaryfeatures. For example, the mounting features on the disc may be pins,and the magnetisation jig/articulated structure have complimentaryfeatures.

Thus, the use of the offset mounting feature pattern shown in FIG. 9enables the disc to be mounted to a complimentary magnetisationjig/articulated structure in a single orientation only.

FIG. 9 also illustrates a further exemplary mounting arrangement. Themounting arrangement comprises one or more alignment notches 61. Whenmounted on the magnetisation jig or articulated structure, the alignmentnotch 61 is aligned with a complimentary feature on the magnetisationjig/articulated structure. This ensures that the disc is in the sameorientation when mounted to both the magnetisation jig and thearticulated structure. FIG. 9 illustrates an alignment notch, but anytype of marking on the disc feature would be suitable, so long as acorresponding marking is provided on the magnetisation jig/articulatedstructure to align the marking of the disc with.

Once the disc has been magnetised during manufacture, the pole positionsmay be accurately measured and recorded. The length of each pole, or theerror of each pole length from the intended length, may be recorded. Theradial distance between the axis of rotation of the disc and the polesmay be recorded. These measurements may be recorded in processing unit14. This characterisation of the pole positions can be subsequently usedwhen the position sensor is in use in order to compensate for errorsintroduced during manufacture. By using a mounting arrangement whichensures the disc is mounted to the articulated structure in the sameorientation that it was mounted to the magnetisation jig, the processingunit is able to map the sensed data from the sensors of the sensorarrays to the recorded characterisation data of the magnetic ring, andcorrect for the known manufacturing errors, thereby resulting in a moreaccurate position measurement.

Alternatively, or additionally, the position sensing arrangement may becalibrated. The calibration process involves generating a correctingfunction which is subsequently applied to position readings in order toproduce corrected position readings which are more accurate. FIG. 10illustrates the magnetic field detected by a magnetic sensor array as amagnetic ring revolves relative to it. 70 denotes the portion of thegraph showing the magnetic field detected from a first magnetic polepair, and 71 denotes the portion of the graph showing the magnetic fielddetected from a second magnetic pole pair. The individual sensors may belocated in positions marked 1, 2, 3 and 4 relative to the magnetic polepairs. In other words, as previously described, the centres of thesensors are separated by a quarter of the length of a pole pair. Thecentres of the outer sensors (marked 1 and 4) are separated by threequarters the length of a pole pair. Theoretically, the magnetic fieldvaries sinusoidally as a magnetic ring moves past the magnetic sensorarray, with one period of the sine wave representing one pole pair.However, imperfections in the manufacturing of the magnetic ring and thealignment of the magnetic sensor assembly with the magnetic ring causesthe magnetic field to deviate from a perfect sine wave.

FIG. 11 illustrates a theoretical sensor reading taken whilst a singlemagnetic pole pair passes the magnetic sensor assembly. This theoreticalsensor reading is multi-bit and represented by line 72 in FIG. 11. FIG.11 also illustrates an actual sensor reading taken whilst a singlemagnetic pole pair passes the magnetic sensor assembly. The actualsensor reading is multi-bit and represented by curve 73 in FIG. 11.

A calibration process will now be described which aims to correctmeasured sensor position readings for the error shown in FIG. 11. Thisprocess is applied to each magnetic ring of the disc individually.

Firstly, a position reading 73 is taken by the magnetic sensor array foreach pole pair of the magnetic ring. This position reading is referredto below as the calibration pole pair position reading for that polepair. For each pole pair, the calibration pole pair position reading 73is compared to a model pole pair position reading 72 in order togenerate a pole pair correcting function for that pole pair. As can beseen from FIG. 11, the curve of the calibration pole pair positionreading 73 oscillates about the straight line of the model pole pairposition reading 72. The curve of the calibration pole pair positionreading 73 may oscillate periodically about the straight line of themodel pole pair position reading 72. For each pole pair, the pole paircorrecting function may be generated by fitting a curve to thecalibration pole pair position reading 73, and then deducting thestraight line of the model pole pair position reading from the fittedcurve. This curve may be fitted using a least squares method.Alternatively, the curve may be fitted using any other method known inthe art. The fitted curve may be described by a periodically oscillatingfunction. For example, the fitted curve may be a sinusoidal function.FIG. 12 illustrates a correcting function for a pole pair, which is asine wave of amplitude A. In other words, Asinθ, where θ is the anglewithin the pole pair sine wave. Although FIG. 12 only shows the firstharmonic, Asinθ, higher harmonics may be included in the correctingfunction for the pole pair.

The pole pair correcting functions of the pole pairs of the magneticring are then averaged to generate an average pole pair correctingfunction for the magnetic ring. If the correcting function of each polepair is represented by a sine wave, then the average pole paircorrecting function is given by:

$\begin{matrix}{\frac{1}{m}{\overset{m}{\sum\limits_{1}}{A_{m}\mspace{14mu} \sin \mspace{14mu} m\; \theta}}} & \left( {{equation}\mspace{14mu} 12} \right)\end{matrix}$

Where m is the number of pole pairs on the magnetic ring.

Suitably, this correcting function is stored in processing unit 14.Subsequently, when the position sensor takes a position measurement, theposition measurement is corrected using the average pole pair correctingfunction. The position measurement comprises a plurality of pole pairposition readings. The average pole pair correcting function is deductedfrom each pole pair position reading of the position measurement,thereby generating a corrected position measurement.

Deducting the average pole pair correcting function from each pole pairposition reading of a magnetic ring is less accurate than utilising theerrors of each individual pole pair. However, storing only one averagepole pair correcting function per magnetic ring reduces the memory usagerequired for the correction. Also, deducting the average pole paircorrecting function is less algorithmically complex than usingindividual pole pair errors, and hence reduces the processing powerrequired for the correction.

FIG. 13 illustrates theoretical sensor readings taken whilst a wholemagnetic ring passes the magnetic sensor assembly. These theoreticalsensor readings are multi-bit and represented by line 74 of FIG. 13.FIG. 13 also illustrates actual sensor readings taken whilst the wholemagnetic ring passes the magnetic sensor assembly. These actual sensorreadings are multi-bit and represented by curve 75 in FIG. 13. The curve76 represents the actual sensor readings which have been corrected forthe pole pair error as described in relation to FIGS. 10 to 12. FIG. 13illustrates a further error in the actual sensor readings. This error isapproximately a sinewave error compared to the theoretical sensorreadings represented by straight line 74.

A calibration process will now be described which aims to correctmeasured sensor position readings for the further error shown in FIG.13. This process is applied to each magnetic ring of the discindividually.

Firstly, a position reading 73 is taken by the magnetic sensor array foreach pole pair of the magnetic ring. As above, this position reading isreferred to as the calibration pole pair position reading for that polepair. For each pole pair, the calibration pole pair position reading 73is compared to a model pole pair position reading 72 in order togenerate a pole pair correcting function for that pole pair as discussedabove. For each pole pair, a corrected calibration pole pair positionreading is then generated by deducting the pole pair correcting functionfrom the calibration pole pair position reading for that pole pair. Inthe example of FIG. 13, curve 76 represents the corrected calibrationpole pair position readings of all the pole pairs of the magnetic ring.

A revolution correcting function is then generated by comparing thecorrected calibration pole pair position readings for all the pole pairson the magnetic ring with model revolution position readings. As can beseen from FIG. 13, the curve of the corrected calibration pole pairposition readings 76 oscillates about the straight line of the modelrevolution position readings 74. The curve of the corrected calibrationpole pair position readings 76 may oscillate periodically about thestraight line of the model revolution position readings 74. Therevolution correcting function may be generated by fitting a curve tothe corrected calibration pole pair position readings 76, and thendeducting the straight line of the model revolution position readings 74from the fitted curve. This curve may be fitted using a least squaresmethod. Alternatively, the curve may be fitted using any other methodknown in the art. The fitted curve may be described by a periodicallyoscillating function. For example, the fitted curve may be a sinusoidalfunction. The sinusoidal function may be a sine wave of amplitude B,i.e. Bsinϕ, where ϕ is the angle within the revolution sine wave.Although in this example, the fitted curve is only a first harmonic,Bsinϕ, higher harmonics may also be included.

The revolution correcting function may be stored in processing unit 14.Subsequently, when the position sensor takes a position measurement, theposition measurement is corrected using the revolution correctingfunction by deducting the revolution correcting function from theposition reading.

Both the described calibration mechanisms may be carried out on theposition measurements, so that the position measurements are correctedusing both the average pole pair correcting function and the revolutioncorrecting function. Alternatively, only one of the calibrationmechanisms may be carried out. This single calibration mechanism may beeither the average pole pair correcting mechanism or the revolutioncorrecting mechanism.

The calibration mechanisms can be carried out once the magnetic disc ismounted to the revolute joint or other elements whose relative rotationare being sensed in the articulated structure. By carrying out thecalibration at this stage, errors introduced during assembly of theposition sensor in place (for example when aligning the magnetic sensorassembly over the magnetic disc) as well as those introduced duringmanufacture may be detected and compensated for. The sensor may bere-calibrated in use, using the calibration mechanisms described above.The calibration mechanisms may be carried out during manufacture, andthe position sensor supplied with the described correcting functions,which are stored in processing unit 14 and subsequently applied tomeasured position readings during use. In the field of robotics, andparticularly surgical robotics, it is desirable for the robot arm to beas small and light as possible. The position sensors used on each jointof the robot arm are also preferably small and light. For example, disc3 may be made of aluminium. In this field, magnetising the disc in-situon the robot arm is impractical. Due to the compact nature of the robotarm, there is insufficient space to apply a standard magnetisation jigaround the robot arm in order to magnetise the disc. Additionally, amagnetic back plate to the disc is used during magnetisation in order tomagnetise the disc. For example, steel is used as this back plate. Inorder to magnetise the disc in-situ, the disc would need to be made ofsteel or another magnetic material. This would preclude making the discof a lightweight material such as aluminium. Thus, the disc is notmagnetised in-situ, but instead the measures described herein are takento replicate the magnetisation jig environment at the articulatedstructure in order to compensate for errors introduced duringmanufacture.

The applicant hereby discloses in isolation each individual featuredescribed herein and any combination of two or more such features, tothe extent that such features or combinations are capable of beingcarried out based on the present specification as a whole in the lightof the common general knowledge of a person skilled in the art,irrespective of whether such features or combinations of features solveany problems disclosed herein, and without limitation to the scope ofthe claims. The applicant indicates that aspects of the presentinvention may consist of any such individual feature or combination offeatures. In view of the foregoing description it will be evident to aperson skilled in the art that various modifications may be made withinthe scope of the invention.

1. A method of assembling a position sensing arrangement configured tosense the position of a revolute joint of an articulated structure, theposition sensing arrangement comprising a magnetic sensor assembly and adisc having a first magnetic ring with j magnetic pole pairs and asecond magnetic ring with k magnetic pole pairs, where |j-k|>1, aboundary of the disc being constrained by the articulated structure, themethod comprising: determining a number of pole pairs of the firstmagnetic ring p such that the first magnetic ring is separated from theconstrained boundary by at least the magnetic sensor assembly;determining a number of pole pairs of the second magnetic ring to be aninteger q such that the second magnetic ring is separated from the firstmagnetic ring by a predetermined distance; and if p and q are co-prime:selecting j to be p and k to be q; and assembling the position sensingarrangement by: mounting the disc to the articulated structure such thatboth the disc and the revolute joint are permitted to rotate about thesame axis; and mounting the magnetic sensor assembly to the articulatedstructure so as to enable detection of relative rotation of the disc andthe magnetic sensor assembly.
 2. A method as claimed in claim 1, whereinif p and q are not co-prime: iteratively determining a further value qfor the number of pole pairs of the second magnetic ring such that thedifference between p and q increments by one each iteration; and foreach iteration, if p and q are co-prime: selecting p to be a bound of arange of numbers of pole pairs of the first magnetic ring, and selectingq to be a bound of a range of numbers of pole pairs of the secondmagnetic ring; identifying one or more other co-prime pair of numbersp′, q′, where: p′ is in the range of numbers of pole pairs of the firstmagnetic ring, and q′ is in the range of numbers of pole pairs of thesecond magnetic ring, and for that p′, q′ pair, the second magnetic ringis separated from the first magnetic ring by at least the predetermineddistance; selecting the identified p′, q′ pair which has the largestvalue of p′; selecting j to be the selected p′ and k to be the selectedq′; and assembling the position sensing arrangement by: mounting thedisc to the articulated structure such that both the disc and therevolute joint are permitted to rotate about the same axis; and mountingthe magnetic sensor assembly to the articulated structure so as toenable detection of relative rotation of the disc and the magneticsensor assembly.
 3. A method as claimed in claim 1, wherein if p and qare not co-prime: iteratively determining a further value q for thenumber of pole pairs of the second magnetic ring such that thedifference between p and q increments by one each iteration; and foreach iteration, if p and q are co-prime: selecting p to be a bound of arange of numbers of pole pairs of the first magnetic ring, and selectingq to be a bound of a range of numbers of pole pairs of the secondmagnetic ring; identifying one or more other co-prime pair of numbersp′, q′, where: p′ is in the range of numbers of pole pairs of the firstmagnetic ring, and q′ is in the range of numbers of pole pairs of thesecond magnetic ring, and for that p′, q′ pair, the second magnetic ringis separated from the first magnetic ring by at least the predetermineddistance; selecting the identified p′, q′ pair which has the largestvalue of q′; selecting j to be the selected p′ and k to be the selectedq′; and assembling the position sensing arrangement by: mounting thedisc to the articulated structure such that both the disc and therevolute joint are permitted to rotate about the same axis; and mountingthe magnetic sensor assembly to the articulated structure so as toenable detection of relative rotation of the disc and the magneticsensor assembly.
 4. A method as claimed in claim 1, wherein if p and qare not co-prime: iteratively determining a further value q for thenumber of pole pairs of the second magnetic ring such that thedifference between p and q increments by one each iteration; and foreach iteration, if p and q are co-prime: selecting p to be a bound of arange of numbers of pole pairs of the first magnetic ring, and selectingq to be a bound of a range of numbers of pole pairs of the secondmagnetic ring; identifying one or more other co-prime pair of numbersp′, q′, where: p′ is in the range of numbers of pole pairs of the firstmagnetic ring, and q′ is in the range of numbers of pole pairs of thesecond magnetic ring, and for that p′, q′ pair, the second magnetic ringis separated from the first magnetic ring by at least the predetermineddistance; selecting the identified p′, q′ pair which has the smallestvalue of q′; selecting j to be the selected p′ and k to be the selectedq′; and assembling the position sensing arrangement by: mounting thedisc to the articulated structure such that both the disc and therevolute joint are permitted to rotate about the same axis; and mountingthe magnetic sensor assembly to the articulated structure so as toenable detection of relative rotation of the disc and the magneticsensor assembly.
 5. A method as claimed in claim 1, wherein if p and qare not co-prime: iteratively determining a further value q for thenumber of pole pairs of the second magnetic ring such that thedifference between p and q increments by one each iteration; and foreach iteration, if p and q are co-prime: selecting p to be a bound of arange of numbers of pole pairs of the first magnetic ring, and selectingq to be a bound of a range of numbers of pole pairs of the secondmagnetic ring; identifying one or more other co-prime pair of numbersp′, q′, where: p′ is in the range of numbers of pole pairs of the firstmagnetic ring and q′ is in the range of numbers of pole pairs of thesecond magnetic ring, and for that p′, q′ pair, the second magnetic ringis separated from the first magnetic ring by at least the predetermineddistance; selecting the identified p′, q′ pair which has the smallestvalue of |p′-q′|; selecting j to be the selected p′ and k to be theselected q′; and assembling the position sensing arrangement by:mounting the disc to the articulated structure such that both the discand the revolute joint are permitted to rotate about the same axis; andmounting the magnetic sensor assembly to the articulated structure so asto enable detection of relative rotation of the disc and the magneticsensor assembly.
 6. A method as claimed in claim 1 wherein the firstmagnetic ring and the second magnetic ring are concentric, the firstmagnetic ring being inside the second magnetic ring, the constrainedboundary being the inner radial boundary of the disc.
 7. A method asclaimed in claim 2, wherein the first magnetic ring and the secondmagnetic ring are concentric, the first magnetic ring being inside thesecond magnetic ring, the constrained boundary being the inner radialboundary of the disc, the method comprising iteratively determining afurther value q by incrementing q by 1 each iteration, wherein p′>p andq′<q, and wherein for each iteration, if p and q are co-prime, themethod comprises selecting p to be the lower bound of a range of numbersof pole pairs of the first magnetic ring, and selecting q to be theupper bound of a range of numbers of pole pairs of the second magneticring.
 8. A method as claimed of claim 1, wherein the first magnetic ringand the second magnetic ring are concentric, the first magnetic ringbeing outside the second magnetic ring, the constrained boundary beingthe outer radial boundary of the disc.
 9. A method as claimed in claim2, wherein the first magnetic ring and the second magnetic ring areconcentric, the first magnetic ring being outside the second magneticring, the constrained boundary being the outer radial boundary of thedisc, the method comprising iteratively determining a further value q bydecrementing q by 1 each iteration, wherein p′<p and q′>q, and whereinfor each iteration, if p and q are co-prime, the method comprisesselecting p to be the upper bound of a range of numbers of pole pairs ofthe first magnetic ring, and selecting q to be the lower bound of arange of numbers of pole pairs of the second magnetic ring.
 10. A methodas claimed in claim 1, wherein if p and q are not co-prime: iterativelydetermining a further value q for the number of pole pairs of the secondmagnetic ring such that the difference between p and q increments by oneeach iteration; and for each iteration, if p and q are co-prime:selecting p to be a bound of a range of numbers of pole pairs of thefirst magnetic ring, and selecting q to be a bound of a range of numbersof pole pairs of the second magnetic ring; identifying one or more otherco-prime pair of numbers p′, q′, where: p′ is in the range of numbers ofpole pairs of the first magnetic ring, and q′ is in the range of numbersof pole pairs of the second magnetic ring, and for that p′, q′ pair, thesecond magnetic ring is separated from the first magnetic ring by atleast the predetermined distance; selecting an identified p′, q′ pairdependent on the maximum angle of rotation of the revolute joint to bedetected by the position sensing arrangement; selecting j to be theselected p′ and k to be the selected q′; and assembling the positionsensing arrangement by: mounting the disc to the articulated structuresuch that both the disc and the revolute joint are permitted to rotateabout the same axis; and mounting the magnetic sensor assembly to thearticulated structure so as to enable detection of relative rotation ofthe disc and magnetic sensor assembly.
 11. A method as claimed in claim1, wherein the disc further comprises a third magnetic ring with 1 polepairs, the method further comprising: determining a number of pole pairsof the third magnetic ring to be an integer s such that the thirdmagnetic ring is separated from the second magnetic ring by a furtherpredetermined distance; and if p, q and s are co-prime: selecting j tobe p, k to be q, and 1 to be s; and assembling the position sensingarrangement by: mounting the disc to the articulated structure such thatboth the disc and the revolute joint are permitted to rotate about thesame axis; and mounting the magnetic sensor assembly to the articulatedstructure so as to enable detection of relative rotation of the disc andthe magnetic sensor assembly.
 12. A position sensing arrangement mountedto an articulated structure comprising: a first magnetic ring having jmagnetic pole pairs; a second magnetic ring having k magnetic polepairs, the second magnetic ring being immovable relative to the firstmagnetic ring; and a magnetic sensor assembly configured to detect therelative position of the magnetic sensor assembly and the first andsecond magnetic rings; wherein j and k are co-prime and |j-k|>1; whereinthe position sensing arrangement is configured to sense position of arevolute joint of the articulated structure, the first magnetic ring andthe second magnetic ring being mounted to the articulated structure suchthat the first magnetic ring, the second magnetic ring and the revolutejoint are all permitted to rotate about the same axis, wherein thesecond magnetic ring is radially separated from the first magnetic ringby a predetermined distance.
 13. A position sensing arrangement asclaimed in claim 12, wherein |j-k|>7.
 14. A position sensing arrangementas claimed in claim 12, wherein the first and second magnetic rings areboth disposed on the same surface of a disc.
 15. (canceled)
 16. Aposition sensing arrangement as claimed in claim 12, wherein the firstand second magnetic rings are disposed on opposing surfaces of a disc.17. (canceled)
 18. A position sensing arrangement as claimed in claim12, wherein the predetermined distance is at least the length of amagnetic pole pair.
 19. (canceled) A position sensing arrangement asclaimed in claim 12, wherein the magnetic sensor assembly comprises afirst magnetic sensor array disposed over the first magnetic ring and asecond magnetic sensor array disposed over the second magnetic ring,adjacent sensors of each of the first and second magnetic sensor arraysbeing separated by a quarter the length of a magnetic pole pair.
 20. Aposition sensing arrangement as claimed in claim 17, wherein the firstmagnetic sensor array has a radial extent less than the radial extent ofthe first magnetic ring, and the second magnetic sensor array has aradial extent less than the radial extent of the second magnetic ring.21. A position sensing arrangement as claimed in claim 17, wherein eachof the first and second magnetic sensor arrays is rectilinear. 22.(canceled)
 23. A position sensing arrangement as claimed in claim 17,wherein at least one of the first and second magnetic sensor arrays isarranged in a circular configuration.
 24. (canceled)